Saxon 5/4

Lesson 21 89
Example 4 Draw a circle with a radius of 2 cm.

Solution Set the compass so that the radius is 2 cm. Place the pivot
point; then swing the pencil point of the compass around it to
draw the circle.

2 cm

The diameter of a circle is the distance across the circle
through the center. As the diagram below illustrates, the
diameter of a circle equals two radii.

radius radius

diameter

Example 5 If the radius of a circle is 2 cm, then what is the diameter of
the circle?

Solution Since the diameter of a circle equals two radii, the diameter
of a circle with a 2-cm radius is 4 cm.

LESSON PRACTICE
Practice set a. Draw a triangle with two sides that are the same length.
b. Draw a rectangle that is about twice as long as it is wide.

c. Use a compass to draw a circle with a radius of 1 inch.

d. What is the diameter of a circle that has a 3-cm radius?

e. What is another name for a rectangle whose length is
equal to its width?

90 Saxon Math 5/4
MIXED PRACTICE

Problem set 1. Hiroshi had four hundred seventeen marbles. Harry had
(1, 13) two hundred twenty-two marbles. How many marbles
did Hiroshi and Harry have in all?

2. Tisha put forty jacks into a pile. After Jane added all of
(11, 14) her jacks there were seventy-two jacks in the pile. How

many jacks did Jane put in?

3. The ones digit is 5. The number is greater than 640 and
(4) less than 650. What is the number?

4. Write seven hundred fifty-three in expanded form.

(16)

5. If x + y = 10, then what is the other addition fact for x, y,
(6) and 10? What are the two subtraction facts for x, y, and 10?

6. The needle is pointing to what 200 400
(18) number on this scale? 0 600

7. Use a centimeter ruler to measure this rectangle.

(Inv. 2)

(a) What is the length?
(b) What is the width?
(c) What is the perimeter?

8. 493 9. $486 10. $524
(13) + 278 (13) + $378 (13) + $109

11. Draw a triangle. Make each side 2 cm long. What is the
(Inv. 2, 21) perimeter of the triangle?

12. Draw a square with sides 2 inches long. What is the
(Inv. 2, 21) perimeter of the square?

Lesson 21 91

13. 17 14. 45 15. 15 16. 62
–(12) A (15) – 29 –(12) B (15) – 45

9 18. 14 6 20. 75
–(16) B (16) – P
17. 24 19. Y
+(14) D 2 (16) – 36 45
22. 14 24. 15
45 (17) 28 53 (17) 24

21. 46 77 23. 14 36
(17) 35 + 23 (17) 23 + 99

27 38
+ 39 + 64

25. Write the next three numbers in each counting sequence:
(3, Inv. 1) (a) …, 28, 35, 42, _____, _____, _____, …

(b) …, 40, 30, 20, _____, _____, _____, …

26. Alba drew a circle with a radius of 4 cm. What was the
(21) diameter of the circle?

A. 8 in. B. 2 in. C. 8 cm D. 2 cm

92 Saxon Math 5/4

LESSON

22 Naming Fractions •

Adding Dollars and Cents

WARM-UP

Facts Practice: 100 Addition Facts (Test A)

Mental Math:

a. 63 + 21 b. 45 + 23 c. 65 + 30
d. 48 + 19 + 200 e. 36 + 29 + 30 f. 130 + 200 + 300

g. What number should be added to each of these numbers for
the total to be 10: 8, 4, 3, 9, 5?

Problem Solving:

The hour hand moves around the face of a clock once in twelve
hours. How many times does the hour hand move around the
face of the clock in a week?

NEW CONCEPTS

Naming Part of a whole can be named with a fraction. A fraction is
fractions written with two numbers. The bottom number of a fraction is
called the denominator. The denominator tells how many equal
parts are in the whole. The top number of a fraction is called the
numerator. The numerator tells how many of the parts are being
counted. When naming a fraction, we name the numerator first;
then we name the denominator using its ordinal number.† Some
fractions and their names are shown below.

1 one half 3 three fifths
2 5

1 one third 5 five sixths
3 6

2 two thirds 7 seven eighths
3 8

1 one fourth 1 one tenth
4 10

†Exception: We use the word half (instead of second) for a denominator of 2.
Also, we may choose to use the word quarter to name a denominator of 4.

Lesson 22 93

Example 1 What fraction of the circle is shaded?

Solution There are four equal parts and three are shaded. Therefore,
the fraction of the circle that is shaded is three fourths, which
we write as

3
4

Example 2 A dime is what fraction of a dollar?
Solution Ten dimes equal one dollar, so one dime is s of a dollar.

Example 3 Three quarters are what fraction of a dollar?
Solution Four quarters equal a dollar, so each quarter is F of a dollar.
Three quarters are H of a dollar.

Adding We add dollars and cents the same way we add whole
dollars and numbers. The dot, called a decimal point, separates dollars
from cents. To add dollars to dollars and cents to cents, we
cents align the decimal points. We remember to write the dollar
sign and the decimal point in the sum.

Example 4 Add: $3.56
+ $2.75

Solution First we add the pennies, then we add the dimes, and then
we add the dollars. Since ten pennies equal a dime and ten
dimes equal a dollar, we regroup when the total in any
column is ten or more.

Add pennies.
Add dimes.
Add dollars.

$3.56
+ $2.75

11

$6.31

94 Saxon Math 5/4
LESSON PRACTICE

Practice set What fraction of each shape is shaded?
a. b.

c. d.

e. A quarter is what fraction of a dollar?

f. A nickel is what fraction of a dollar?

g. Three dimes are what fraction of a dollar?

Add: i. $3.65
h. $2.75 + $4.28

+ $2.75

MIXED PRACTICE

Problem set 1. The first four odd numbers are 1, 3, 5, and 7. What is
(1) their sum?

2. James was 49 inches tall at the beginning of summer. He
(1, 9) grew 2 inches over the summer. How tall was James at

the end of summer?

3. Use the digits 1, 2, and 3 once each to write an odd
(10) number less than 200.

Write the next three numbers in each counting sequence:
4. …, 80, 72, 64, _____, _____, _____, …

(3)

5. …, 60, 54, 48, _____, _____, _____, …

(3)

6. Draw a square with sides 3 cm long. What is the
(Inv. 2, 21) perimeter of the square?

7. A yardstick is how many feet long?

(Inv. 2)

Lesson 22 95

8. What is the place value of the 9 in 891?

(4)

9. Write 106 in expanded form. Then use words to write
(7, 16) the number.

10. Use the numbers 6, 9, and 15 to write two addition facts
(6) and two subtraction facts.

11. Use digits and symbols to write that eighteen is greater
(Inv. 1) than negative twenty.

12. (a) Round 28 to the nearest ten.
(20)
(b) Round $5.95 to the nearest dollar.

13. A bicycle is about how many meters long?

(Inv. 2)

14. The needle is pointing to what 100

(18) number on this scale? 80 120

15. Draw a circle that has a diameter of 2 centimeters. What
(21) is the radius of the circle?

16. What fraction of this rectangle
(22) is shaded?

17. The door was two meters tall. Two meters is how many
(Inv. 2) centimeters?

18. 51 – 43 19. 70 – 44 20. 37 – 9
(15) (15) (15)

21. $8.79 + $0.64 22. $5.75 + $2.75
(22) (22)

23. N 24. X 25. 37
(14) + 13 (16) – 42 –(16) P

17 27 14

26. A number sentence such as 20 + N = 60 can be called
(6) an equation. If this equation is true, then which of the
following equations is not true?

A. 60 – 20 = N B. 60 – N = 20

C. N – 20 = 60 D. N + 20 = 60

96 Saxon Math 5/4

LESSON

23 Lines, Segments,

Rays, and Angles

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math: e. 86 f. 360
Add hundreds, then tens, and then ones: + 210 25

a. 320 b. 645 c. 145 d. 632 + 214
+ 256 + 32 + 250 + 55

g. What number should be added to each of these numbers for
the total to be 10: 2, 6, 7, 1, 5?

Patterns:

On a hundred number chart, shade the squares that contain a
multiple of 9. Then write the numbers in the shaded squares
from 9 to 90 in a column. What patterns can you find in the
column of numbers?

NEW CONCEPT
A line goes on and on. When we draw a line, we include an
arrowhead on each end to show that the line continues in
both directions.

Line

Part of a line is a line segment, or just segment. When we
draw a segment, we do not include arrowheads. We can,
however, use dots to show the endpoints of the segment.

Segment

A ray is sometimes called a half line. A ray begins at a point
and continues in one direction without end. When we draw a
ray, we include an arrowhead on one end.

Ray

Lesson 23 97

Example 1 Write “line,” “segment,” or “ray” to describe each of these
physical models:
(a) a beam of starlight
(b) a ruler

Solution (a) A beam of starlight begins at a “point,” the star, and
continues across space. This is an example of a ray.

(b) A ruler has two endpoints, so it is best described as an
example of a segment.

Lines and segments that go in the same direction and stay the
same distance apart are parallel.

Pairs of parallel lines and segments

When lines or segments cross, we say they intersect.

Pairs of intersecting lines and segments

Intersecting lines or segments that form “square corners” are
perpendicular.

Pairs of perpendicular lines and segments

Angles are formed where lines or segments intersect or where
two or more rays or segments begin. An angle has a vertex and
two sides. The vertex is the point where the two sides meet (the
“corner”).

side

vertex side

98 Saxon Math 5/4
An angle is named by how “open” it is. An angle like the
corner of a square is called a right angle.

Square Right angles

To show that an angle is a right angle, we can draw a small
square in the corner of the right angle.

This mark shows that the
angle is a right angle.

Angles that are smaller than right angles are called acute angles.
Some people remember this by saying, “a cute little angle.”
Angles that are larger than right angles are obtuse angles.

Acute angle Obtuse angle

Example 2 Describe each of these figures as an acute, obtuse, or right angle.
(a) (b) (c)

Solution (a) The angle is smaller than a right angle, so it is an
acute angle.

(b) The angle makes a square corner, so it is a right angle.
(c) The angle is larger than a right angle, so it is an

obtuse angle.

The figure in the following example has four angles. We can
name each angle by the letter at the vertex of the angle. The four
angles in the figure are angle Q, angle R, angle S, and angle T.

Example 3 Describe each of the four angles in this T Q
figure as acute, right, or obtuse.

SR

Solution Angle Q is acute. Angle R is obtuse. Angles S and T are right
angles.

Lesson 23 99
Example 4 Draw a triangle that has one right angle.

Solution We begin by drawing two line segments that form a right
angle. Then we draw the third side.

Notice that the other two angles are acute angles.
Activity: Real-World Segments and Angles

1. Look for examples of the following figures in your
classroom. List the examples on the board.
(a) parallel segments
(b) perpendicular segments
(c) right angles
(d) acute angles
(e) obtuse angles

2. Bend your arm so that the angle at the elbow is an acute
angle, then a right angle, then an obtuse angle. Bend your
leg so that the angle at the knee is an acute angle, then a
right angle, then an obtuse angle.

LESSON PRACTICE
Practice set a. Draw two segments that intersect but are not perpendicular.
b. Draw two lines that are perpendicular.
c. Draw a ray.
d. Are the rails of a train track parallel or perpendicular?
e. A triangle has how many angles?
f. Which of these angles does not look like a right angle?
A. B. C.

100 Saxon Math 5/4
MIXED PRACTICE

Problem set 1. Twenty-eight children were in the first line. Forty-two
(1, 9) children were in the second line. Altogether, how many
children were in both lines?

2. Tina knew that there were 28 books in the two stacks.
(11, 14) Tina counted 12 books in the first stack. Then she figured

out how many books were in the second stack. How
many books were in the second stack?

3. Use the digits 1, 2, and 3 once each to write an odd
(10) number greater than 300.

4. Write the next three numbers in each counting sequence:

(3)

(a) …, 40, 36, 32, _____, _____, _____, …
(b) …, 30, 27, 24, _____, _____, _____, …

5. Use the numbers 15, 16, and 31 to write two addition
(6) facts and two subtraction facts.

6. Use digits and a comparison symbol to write that six
(Inv. 1) hundred thirty-eight is less than six hundred eighty-three.

7. (a) Round 92 to the nearest ten.

(20)

(b) Round $19.67 to the nearest dollar.

8. The diameter of a nickel is 2 centimeters. If 10 nickels are
(Inv. 2, 21) placed in a row, how long will the row be? Count by twos.

9. Use a centimeter ruler to measure this rectangle:

(Inv. 2)

(a) What is the length?
(b) What is the width?
(c) What is the perimeter?

10. Which of these shapes has four right angles?
(23)
A. B. C.

Lesson 23 101

11. What fraction of this triangle
(22) is shaded?

12. It is afternoon. What time is shown 11 12 1
(19) on this clock? 10 2

9 3

8 4
76 5

13. $83 14. 42 15. 72 16. $4.28
(15) – $27 (15) – 27 (15) – 36 (22) + $1.96

17. $4.36 18. 57 19. 67 20. K
(22) + $2.95 (14) + K –(16) B (16) – 22

88 16 22

21. 42 – 7 22. 55 – 48 23. 31 – 20
(15) (15) (14)

24. 25 + 25 + 25 + 25
(17)

25. (a) How many nickels equal one dollar?
(22)
(b) One nickel is what fraction of a dollar?

(c) Seven nickels are what fraction of a dollar?

26. If 26 + M = 63, then which of these equations is not true?
(6)
A. M + 26 = 63 B. M – 63 = 26

C. 63 – M = 26 D. 63 – 26 = M

27. Which of these figures illustrates a ray?
(23)
A.

B.

C.

102 Saxon Math 5/4

LESSON

24 More About Missing Numbers

in Addition and Subtraction

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Add hundreds, then tens, and then ones:

a. 365 + 321 b. 650 + 45 c. 40 + 300 + 25

d. 500 + 40 + 16 e. 300 + 50 + 12 f. 400 + 80 + 11

g. Seven can be split into 3 + 4. If seven is split into
2 + , what number is represented by ?

Problem Solving:

The pair of numbers 1 and 8 have the sum of 9. List three more
pairs of counting numbers that have a sum of 9.

NEW CONCEPT

We have seen that the three numbers in an addition or
subtraction fact form three other facts as well. If we know
that N + 5 = 14, then we know these four facts:

N 5 14 14
+N –N –5
+5
14 5 N
14

Notice that the last of these facts, 14 – 5 = N, shows us how
to find N. We subtract 5 from 14 to find that N equals 9.

Example 1 Write another addition fact and two subtraction facts using
the numbers in this equation:

36 + M = 54

Which fact shows how to find M?

Solution We arrange the numbers to write three facts. Notice that the
sum, 54, becomes the first number of both subtraction facts.

M + 36 = 54 54 – M = 36 54 – 36 = M
The fact that shows how to find M is

54 – 36 = M

Lesson 24 103

Example 2 Write another subtraction fact and two addition facts using
the numbers in this equation:
72 – W = 47
Which fact shows how to find W?

Solution Notice that the first number of a subtraction fact remains the
first number of the second subtraction fact.
72 – 47 = W
Also notice that the first number of a subtraction fact is the
sum when the numbers are arranged to form an addition fact.
47 + W = 72 W + 47 = 72
The fact that shows how to find W is
72 – 47 = W

Example 3 Find the missing number: R + 36 = 54

Solution We can form another addition fact and two subtraction facts
using these numbers.
36 + R = 54 54 – R = 36 54 – 36 = R
The last fact, 54 – 36 = R, shows us how to find R. We
subtract 36 from 54 and get 18.

Example 4 Find the missing number: T – 29 = 57

Solution We can write the first number of a subtraction equation as the
sum of an addition equation.
57 + 29 = T
Thus, T equals 86.

LESSON PRACTICE b. Q + 17 = 45
Practice set Find each missing number: d. N – 26 = 68
a. 23 + M = 42 f. 62 – A = 26

c. 53 – W = 28

e. 36 + Y = 63

104 Saxon Math 5/4

MIXED PRACTICE

Problem set 1. Rafael placed two 1-foot rulers end to end. What was the
(Inv. 2) total length of the two rulers in inches?

2. There were 47 apples in the big tree. There was a total of
(11, 24) 82 apples in the big tree and in the little tree. How many

apples were in the little tree?

3. All the students lined up in two equal rows. Which could
(10) not be the total number of students?

A. 36 B. 45 C. 60

4. Find the missing numbers in this counting sequence:

(3)

…, 9, 18, _____, _____, 45, _____, …

5. Find the sixth number in this counting sequence:

(3, 5)

7, 14, 21, …

6. Compare: 15 – 9 À 13 – 8

(Inv. 1)

7. (a) Round 77 to the nearest ten.

(20)

(b) Round $29.39 to the nearest dollar.

8. A professional basketball player might be about how
(Inv. 2) many meters tall?

9. It is morning. What time is shown 11 12 1
(19) on this clock? 10 2

9 3

8 4
76 5

10. Which street is parallel to Elm? ELM
(23) OAK

BROADWAY

11. (a) How many dimes equal one dollar?
(22)
(b) One dime is what fraction of a dollar?
(c) Nine dimes are what fraction of a dollar?

Lesson 24 105
12. Draw a rectangle that is 5 centimeters long and
(Inv. 2, 21) 2 centimeters wide. What is the perimeter?
13. Name each type of angle shown below.
(23) (a) (b) (c)

14. $31 15. $468 16. 57 17. $4.97
(15) – $14 (13) + $247 (14) – 37 (22) + $2.58

18. 36 – C = 19 19. B + 65 = 82
(24) (24)

20. 87 + D = 93 21. N – 32 = 19
(24) (24)

22. 48 – 28 23. 41 – 32 24. 76 – 58
(14) (15) (15)

25. 416 + 35 + 27 + 43 + 5
(17)

26. Which point on this number line could represent –3?
(Inv. 1)
w x yz

–10 0 10

A. point w B. point x C. point y D. point z

27. Describe how a segment is different from a line.
(23)

106 Saxon Math 5/4

LESSON

25 Subtraction Stories

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Add hundreds, then tens, and then ones:

a. 340 + 50 + 200 b. 200 + 50 + 432 c. 560 + 200 + 25

Review:

d. 56 + 19 + 200 e. 48 + 39 + 100 f. 36 + 9

g. Complete each split: 6 = 2 + 6=3+

Problem Solving:

If the sun rose at 5:00 a.m. and set at 7:00 p.m., how many hours
of sunlight were there?

NEW CONCEPT

We have practiced “some and some more” story problems.
“Some and some more” stories have an addition pattern.

In this lesson we will begin practicing story problems that
have a subtraction pattern. One type of story with a
subtraction pattern is a “some went away” story. Read this
“some went away” story:

John had 7 marbles. Then he lost 3 marbles.
He has 4 marbles left.

We can write the information from this story in a subtraction
pattern like this:

PATTERN PROBLEM
Some
– Some went away 7 marbles
What is left – 3 marbles

4 marbles

We can also write the pattern sideways.

PATTERN: Some – some went away = what is left

PROBLEM: 7 marbles – 3 marbles = 4 marbles

Lesson 25 107

In a “some went away” story there are three numbers. Any
one of the numbers could be missing. We write the numbers
in a subtraction pattern and then find the missing number. A
diagram may help us understand the action in a “some went
away” story.

Began with ‡

Some What
went is
away left
›
‹

Example 1 Jimmy had some marbles. Then he lost 15 marbles. Now he
has 22 marbles left. How many marbles did Jimmy have in
the beginning?

Solution Jimmy lost some marbles. This story has a subtraction
pattern. We are told how many marbles “went away” and
how many marbles are left. To find how many marbles Jimmy
had in the beginning, we write the numbers in a subtraction
pattern and use a letter for the missing number.

PATTERN PROBLEM
Some M marbles
– Some went away – 15 marbles
What is left 22 marbles

We can find the missing number in this subtraction problem
by adding.

22 marbles
+ 15 marbles

37 marbles

Jimmy had 37 marbles in the beginning. Now we check the
answer.

37 marbles
– 15 marbles

22 marbles check

Example 2 Celia had 42 marbles. She lost some marbles. She has
29 marbles left. How many marbles did Celia lose?

108 Saxon Math 5/4

Solution Celia lost some marbles. This story has a subtraction pattern,
and we want to find the number that went away. We write the
numbers in the pattern.

PATTERN PROBLEM
Some 42 marbles
– Some went away – M marbles
What is left 29 marbles

To find the missing number, we subtract.

42
– 29

13

We find that Celia lost 13 marbles. Now let’s see whether
13 marbles makes the pattern correct.

42 marbles
– 13 marbles

29 marbles check

Example 3 Fatima had 65 marbles. Then she lost 13 marbles. How many
marbles does Fatima have left?

Solution Again we have a subtraction story. We write the numbers in a
subtraction pattern and then find the missing number. This
time, we practice writing the pattern sideways.
PATTERN: Some – some went away = what is left
PROBLEM: 65 marbles – 13 marbles = M marbles
To find the missing number, we simply subtract.
65 marbles – 13 marbles = 52 marbles
We find that Fatima has 52 marbles left.

LESSON PRACTICE

Practice set For each problem, write a subtraction pattern. Then answer
the question.

a. Marko had 42 marbles. Then he lost some marbles. Now
he has 26 marbles. How many marbles did Marko lose?

b. Tamika lost 42 marbles. Now she has 26 marbles. How
many marbles did Tamika have in the beginning?

c. Barbara had 75 cents. Then she spent 27 cents. How
many cents does Barbara have now?

Lesson 25 109

MIXED PRACTICE

Problem set 1. Micky had 75 rocks. Then she lost some rocks. Now she
(25) has 27 rocks. How many rocks did Micky lose? Write a

subtraction pattern and solve the problem.

2. Sixty-three birds sat in the tree. Then fourteen birds flew
(25) away. How many birds remained in the tree? Write a

subtraction pattern and solve the problem.

3. There were many cats in the alley at noon. Seventy-five
(25) cats ran away. Forty-seven cats remained. How many cats

were in the alley at noon? Write a subtraction pattern and
solve the problem.

4. There are 12 months in a whole year. How many months
(5) are in half of a year?

5. Find the missing numbers in each counting sequence:
(3, Inv. 1) (a) …, 5, 10, _____, _____, 25, _____, …

(b) …, 5, 0, _____, _____, –15, _____, …

6. Use digits and a comparison symbol to write that seven
(Inv. 1) hundred sixty-two is less than eight hundred twenty-six.

7. (a) Round 78 to the nearest ten.

(20)

(b) Round $7.80 to the nearest dollar.

8. If the diameter of a wheel on Joshua’s bike is 20 inches,
(21) then what is the radius of the wheel?

9. It is afternoon. What time is shown 11 12 1
(19) on this clock? 10 2

9 3

8 4
76 5

10. Which street is perpendicular ELM BROADWAY
(23) to Elm? OAK

110 Saxon Math 5/4
11. What fraction of this shape is shaded?

(22)

12. Draw a square whose sides are 4 cm long. What is the
(Inv. 2, 21) perimeter of the square?

13. To what number is the arrow pointing?

(Inv. 1)

270 280 290

14. $52 15. 476 16. 62 17. $4.97
(15) – $14 (13) + 177 (15) – 38 (22) + $2.03

18. 36 19. 55 20. D 21. Y
(24) – G +(24) B (24) – 23 (24) + 14

18 87 58 32

22. 42 – 37 23. 52 – 22
(15) (14)

24. 73 – 59 25. 900 + 90 + 9
(15) (17)

26. Which of these is not equivalent to one meter?
(Inv. 2)
A. 1000 mm B. 100 cm C. 1000 km

27. Describe how a ray is different from a segment.
(23)

Lesson 26 111

LESSON

26 Drawing Pictures of Fractions

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Add from the left and then regroup ones. For example, 35 + 26 is
50 plus 11, which is 61.

a. 55 b. 36 c. 48 d. 37 e. 235 f. 156
+ 26 + 22 + 45 + 145 + 326
+ 25

g. Complete each split: 8 = 1 + 8=3+

Problem Solving:

Jennifer has three coins in her left pocket that total 65¢. What
coins does Jennifer have in her left pocket?

NEW CONCEPT
We can understand fractions better if we learn to draw
pictures that represent fractions.

Example 1 Draw a rectangle and shade two thirds of it.
Solution On the left, we draw a rectangle. Then we divide the
rectangle into three equal parts. As a final step, we shade any
two of the equal parts.

Rectangle 3 equal parts 2 parts shaded

There are other ways to divide the rectangle into three equal
parts. Here is another way we could shade two thirds of the
rectangle:

Rectangle 3 equal parts 2 parts shaded

112 Saxon Math 5/4
Example 2 Draw a circle and shade one fourth of it.
Solution First we draw a circle. Then we divide the circle into four
equal parts. Then we shade any one of the parts.

Circle 4 equal parts 1 part shaded

LESSON PRACTICE
Practice set a. Draw a square and shade one half of it.

b. Draw a rectangle and shade one third of it.

c. Draw a circle and shade three fourths of it.

d. Draw a circle and shade two thirds of it.

e. Is one half of this circle shaded?
Why or why not?

MIXED PRACTICE

Problem set 1. Mary had 42 pebbles. She threw some into the lake. Then
(25) she had 27 pebbles left. How many pebbles did Mary
0
throw into the lake? Write a subtraction pattern and solve
the problem.

2. Demosthenes had a bag of pebbles when the sun came
(25) up. He put 17 pebbles in his mouth. Then there were

46 pebbles left in the bag. How many pebbles were in
the bag when the sun came up? Write a subtraction
pattern and solve the problem.

3. Franklin saw one hundred twelve stars. Eleanor looked the
00 (11, 13) other way and saw some more stars. If they saw three

hundred seventeen stars in all, how many did Eleanor see?
Write an addition pattern and solve the problem.

Lesson 26 113
4. Use the digits 4, 5, and 6 once each to write an even
(10) number less than 500.

5. Draw a square and shade three fourths of it.

(26)

6. What is the perimeter of this 10 cm 6 cm
(Inv. 2) triangle?

8 cm

7. Use digits and symbols to show that negative twenty is
(Inv. 1) less than negative twelve.

8. (a) Round 19 to the nearest ten.

(20)

(b) Round $10.90 to the nearest dollar.

9. One meter equals how many centimeters?

(Inv. 2)

10. It is before noon. What time is 11 12 1
(19) shown on this clock? 10 2

9 3

8 4
76 5

11. Which street makes a right angle ELM BROADWAY
(23) with Oak? OAK

12. What fraction of this figure is
(22) shaded?

13. This scale shows weight in 300 400
(18) pounds. What number of pounds is 200 500

the needle pointing to?

114 Saxon Math 5/4

14. Y 15. $486 16. $68 17. $5.97
(24) + 63 (13) + $277 (15) – $39 (22) + $2.38

81

18. N + 42 = 71 19. 87 – N = 65
(24) (24)

20. 27 + C = 48 21. E – 14 = 28
(24) (24)

22. 42 – 29 23. 77 – 37 24. 41 – 19
(15) (14) (15)

25. 4 + 7 + 15 + 21 + 5 + 4 + 3
(17)

26. In which figure is A not shaded? D.
(26)
A. B. C.

27. Is the largest angle of this triangle acute, right, or obtuse?
(23)

Lesson 27 115

LESSON

27 Multiplication as Repeated

Addition • Elapsed Time

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Add hundreds, then tens, and then ones; regroup the ones:

a. 25 + 36 b. 147 + 225 c. 30 + 25 + 26

d. 356 + 26 e. 46 + 10 + 28 f. 350 + 35 + 7

g. Complete each split: 9 = 3 + 9=4+

Problem Solving:

The two-digit numbers 18 and 81 are written with digits whose
sum is nine. On your paper, list in order the two-digit numbers
from 18 to 81 whose digits have a sum of nine.

18, ___, ___, ___, ___, ___, ___, 81

NEW CONCEPTS
Multiplication Suppose we want to find the total number of dots shown on

as repeated these four dot cubes:
addition

One way we can find the total number of dots is to count
the dots one by one. Another way is to recognize that there
are 5 dots in each group and that there are four groups. We
can find the answer by adding four 5’s.

5 + 5 + 5 + 5 = 20

We can also use multiplication to show that we want to add 5
four times.

4 ¥ 5 = 20 or 5

¥4

20

If we find the answer this way, we are multiplying. We call the
¥ a multiplication sign. We read 4 ¥ 5 as “four times five.”

116 Saxon Math 5/4

Example 1 Change this addition problem to a multiplication problem:
6+6+6+6+6

Solution We see five 6’s. We can change this addition problem to a
multiplication problem by writing either

5 ¥ 6 or 6
¥5

Elapsed time The amount of time between two different clock times is
called elapsed time. We can count forward or backward on a
clock to solve elapsed-time problems.

Example 2 If it is afternoon, what time will it be in 11 12 1
3 hours and 20 minutes? 10 2

Solution First we count forward on the clock 9 3
face 20 minutes. From that point, we
count forward 3 hours. 8 4
76 5

Step 1: Counting forward 20 minutes from 1:45 p.m. makes it
2:05 p.m.

Step 2: Counting forward 3 hours from 2:05 p.m. makes it
5:05 p.m.

Example 3 If it is afternoon, what time was it 4 hours 11 12 1
and 25 minutes ago? 10 2

Solution First we count back the number of 9 3
minutes. Then we count back the
number of hours. 8 4
76 5

Step 1: Counting back 25 minutes from 1:15 p.m. makes it
12:50 p.m.

Step 2: Counting back 4 hours from 12:50 p.m. makes it
8:50 a.m.

LESSON PRACTICE

Practice set Change each addition problem to a multiplication problem:

a. 3 + 3 + 3 + 3 b. 9 + 9 + 9

c. 7 + 7 + 7 + 7 + 7 + 7

d. 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

Lesson 27 117

Use the clock to answer problems e and f. 11 12 1
e. If it is morning, what time will it be 10 2
in 2 hours and 25 minutes?
9 3
f. If it is morning, what time was it
6 hours and 30 minutes ago? 8 4
76 5

MIXED PRACTICE

Problem set 1. Just before high noon Adriana saw seventy-eight kittens
(25) playing in the field. At high noon she saw only forty-two
0
kittens playing in the field. How many kittens had left the
field by high noon? Write a subtraction pattern and solve
the problem.

2. If each side of a square floor tile is one foot long, then
(Inv. 2, 21) (a) each side is how many inches long?

(b) the perimeter of the tile is how many inches?

3. List the even numbers between 31 and 39.

(10)

Find the next three numbers in each counting sequence:
4. …, 12, 15, 18, _____, _____, _____, …

(3)

5. …, 12, 24, 36, _____, _____, _____, …

(3)

6. Write 265 in expanded form.

(16)

7. Use words to write –19.

(Inv. 1)

8. (a) Round 63 to the nearest ten.

(20)

(b) Round $6.30 to the nearest dollar.

9. Compare: (b) –15 À –20
(Inv. 1) (a) 392 À 329

10. To what number is the arrow pointing?

(Inv. 1)

200 400 600 800

118 Saxon Math 5/4
11. Draw a square with sides 2 centimeters long. Then shade

(21, 26) one fourth of the square.
12. What fraction of this figure is
(22) shaded?

13. It is afternoon. What time will it be 11 12 1
(27) 3 hours from now? 10 2

9 3

8 4
76 5

14. $67 15. 483 16. 71 17. $5.88
(15) – $29 (13) + 378 (15) – 39 (22) + $2.39

18. D 19. 66 20. 87 21. B
(24) + 19 +(24) F –(24) R (24) – 14

36 87 67 27

22. 400 – 300 23. 663 – 363
(14) (14)

24. Change this addition problem to a multiplication problem:
(27)
9+9+9+9

25. (a) How many pennies equal one dollar?
(22)
(b) A penny is what fraction of a dollar?
(c) Eleven pennies are what fraction of a dollar?

26. If ó = 3 and ∆ = 4, then what does ó + ∆ + ó equal?
(1)
A. 343 B. 7 C. 10 D. 11

27. Draw a dot on your paper to represent a point, and from
(23) that point draw two perpendicular rays.

Lesson 28 119

LESSON

28 Multiplication Table

WARM-UP

Facts Practice: 100 Addition Facts (Test A)

Mental Math:

a. 54 + 36 b. 54 + 19 c. 54 + 120

d. 350 + 30 + 200 e. 210 + 25 + 35 f. 48 + 29

g. Complete each split: 5 = 1 + 5=3+

Problem Solving:

At 12:00 the hands of the clock point in the same direction. At
6:00 the hands point in opposite directions. At what hours do the
hands of a clock form right angles?

NEW CONCEPT

Here we show sequences for counting by ones and twos:

Ones: 1 2 3 4 5 6 7 8 9 10 11 12
Twos: 2 4 6 8 10 12 14 16 18 20 22 24

These sequences—and those for threes and fours and so on
through twelves—appear in this multiplication table:

Multiplication Table

0 1 2 3 4 5 6 7 8 9 10 11 12
0 0000000000000
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 14 16 18 20 22 24
3 0 3 6 9 12 15 18 21 24 27 30 33 36
4 0 4 8 12 16 20 24 28 32 36 40 44 48
5 0 5 10 15 20 25 30 35 40 45 50 55 60
6 0 6 12 18 24 30 36 42 48 54 60 66 72
7 0 7 14 21 28 35 42 49 56 63 70 77 84
8 0 8 16 24 32 40 48 56 64 72 80 88 96
9 0 9 18 27 36 45 54 63 72 81 90 99 108
10 0 10 20 30 40 50 60 70 80 90 100 110 120
11 0 11 22 33 44 55 66 77 88 99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144

120 Saxon Math 5/4

From a multiplication table we can find the answer to
problems such as 3 ¥ 4 by using rows and columns. Rows
run left to right, and columns run top to bottom. We start by
finding the row that begins with 3 and the column that begins
with 4. Then we look for the number where the row and
column meet.

Column

0 1 2 3 4 5 6 7 8 9 10 11 12
0 0000000000000
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 14 16 18 20 22 24
Row 3 0 3 6 9 12 15 18 21 24 27 30 33 36
4 0 4 8 12 16 20 24 28 32 36 40 44 48
5 0 5 10 15 20 25 30 35 40 45 50 55 60
6 0 6 12 18 24 30 36 42 48 54 60 66 72
7 0 7 14 21 28 35 42 49 56 63 70 77 84
8 0 8 16 24 32 40 48 56 64 72 80 88 96
9 0 9 18 27 36 45 54 63 72 81 90 99 108
10 0 10 20 30 40 50 60 70 80 90 100 110 120
11 0 11 22 33 44 55 66 77 88 99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144

Each of the two numbers multiplied is called a factor. The
answer to a multiplication problem is called a product. In
this problem, 3 and 4 are factors, and 12 is the product.
Now look at the row that begins with 4 and the column
that begins with 3. We see that the product of 4 and 3 is
also 12. Changing the order of factors does not change the
product. This is true for any two numbers that are
multiplied and is called the commutative property of
multiplication.

Here are two more properties of multiplication we can see in
the multiplication table. Notice that the product of zero and
any number is zero. This is called the property of zero for
multiplication. Also notice that the product of 1 and any
other factor is the other factor. This is called the identity
property of multiplication.

Lesson 28 121

The three properties we have looked at are summarized in
this table. The letters M and N can be any two numbers. Later,
we will learn about two other properties of multiplication.

Properties of Multiplication

Commutative M¥N=N¥M
property

Identity 1¥N=N
property

Zero 0¥N=0
property

LESSON PRACTICE

Practice set Use the multiplication table to find each product:

a. 9 b. 3 c. 6 d. 4
¥3 ¥9 ¥4 ¥6

e. 7 f. 8 g. 5 h. 8
¥8 ¥7 ¥8 ¥5

i. 10 j. 10 k. 11 l. 12
¥ 10 ¥8 ¥9 ¥ 12

m. Which property of multiplication is shown below?
12 ¥ 11 = 11 ¥ 12

n. Use the zero property of multiplication to find the product:
0 ¥ 25

o. Use the identity property of multiplication to find the
product:
1 ¥ 25

MIXED PRACTICE
Problem set 1. Hansel ate seventy-two pieces of gingerbread. Gretel ate
(1, 17) forty-two pieces of gingerbread. How many pieces of
gingerbread did they eat in all?

2. Sherri needs $35 to buy a baseball glove. She has saved
(11, 24) $18. How much more money does she need?

122 Saxon Math 5/4

3. Draw a rectangle that is 4 cm long and 3 cm wide. What is
(Inv. 2, 21) the perimeter of the rectangle?

Find the missing numbers in each counting sequence:
4. …, 12, _____, _____, 30, 36, _____, …

(3)

5. …, 36, _____, _____, 24, 20, _____, …

(3)

6. Change this addition problem to a multiplication
(27, 28) problem. Then find the product on the multiplication

table shown in this lesson.
6+6+6+6+6+6+6

7. (a) Round 28 to the nearest ten.

(20)

(b) Round $12.29 to the nearest dollar.

8. A right triangle has one right angle. Draw a right
(Inv. 2, 23) triangle. Draw the two perpendicular sides 3 cm long

and 4 cm long.

9. It is morning. What time will it be 11 12 1
(27) 10 minutes from now? 10 2

9 3

8 4
76 5

10. What fraction of this group is
(22) shaded?

11. Write 417 in expanded form. Then use words to write the
(7, 16) number.

12. What temperature is shown on this 10°C
(18) thermometer?

0°C

–10°C

Lesson 28 123

13. 76 14. $286 15. $73 16. $5.87
(15) – 29 (13) + $388 (15) – $39 (22) + $2.43

17. 46 – C = 19 18. N + 48 = 87
(24) (24)

19. 29 + Y = 57 20. D – 14 = 37
(24) (24)

21. 78 – 43 22. 77 – 17 23. 53 – 19
(14) (14) (15)

24. Use the multiplication table to find each product:
(28) (a) 8 ¥ 11
(b) 7 ¥ 10 (c) 5 ¥ 12

25. Compare: 1 yard À 1 meter

(Inv. 1, Inv. 2)

26. Which of the following shows 3 ones and 4 hundreds?
(4)
A. 304 B. 403 C. 4003 D. 3400

27. The product of 9 and 3 is 27. How many times does this
(28) product appear in this lesson’s multiplication table? What

property of multiplication does this show?

124 Saxon Math 5/4

LESSON

29 Multiplication Facts

(0’s, 1’s, 2’s, 5’s)

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

We can split numbers to help us add. Adding 35 and 8, we may
notice that 35 needs 5 more to make 40, and that 8 splits into
5 + 3. So to add 35 and 8, we could add 35 + 5 + 3.

a. 35 + 7 b. 26 + 8 c. 38 + 5
d. 47 + 6 e. 68 + 7 f. 45 + 8

Problem Solving:

Hope has seven coins in her right pocket. None of the coins are
dollar or half-dollar coins. What is the lowest possible value of all
seven coins? What is the highest possible value of all seven coins?

NEW CONCEPT

We will begin memorizing the basic multiplication facts.
Eighty-eight of the facts in the multiplication table shown in
Lesson 28 have 0, 1, 2, or 5 as one of the factors. These facts
are easy to learn.

Zero times any number equals zero.

0¥5=0 5¥0=0 7¥0=0 0¥7=0

One times any number equals the number.

1¥5=5 5¥1=5 7¥1=7 1¥7=7

Two times any number doubles the number.

2 ¥ 5 = 10 2 ¥ 7 = 14 2 ¥ 6 = 12 2 ¥ 8 = 16

Five times any number equals a
number that ends in zero or in five.

5¥1=5 5 ¥ 3 = 15 5 ¥ 7 = 35 5 ¥ 8 = 40

Until we have memorized the facts, we can find multiples of
2 by counting by twos. So 6 ¥ 2 is the sixth number we say
when counting by twos: 2, 4, 6, 8, 10, 12. We can find
multiples of 5 by counting by fives. The sixth number we say
when counting by fives is 30, so 6 ¥ 5 = 30. However,
counting is not a substitute for memorizing the facts.

Lesson 29 125

LESSON PRACTICE
Practice set Take Facts Practice Test C
(Multiplication Facts: 0’s, 1’s, 2’s, 5’s).

MIXED PRACTICE

Problem set 1. Ninety-two blackbirds squawked noisily in the tree. Then
(25) some flew away. Twenty-four blackbirds remained. How
0
many blackbirds flew away? Write a subtraction pattern
and solve the problem.

2. Robill collected 42 seashells. Then Buray collected some
(11, 24) seashells. They collected 83 seashells in all. How many

seashells did Buray collect?

3. Conner estimated that the radius of one of the circles on
(Inv. 2, 21) the playground was 2 yards. If Conner was correct, then

(a) the radius was how many feet?

(b) the diameter was how many feet?

Find the missing numbers in each counting sequence:
4. …, 8, _____, _____, 32, 40, _____, …

(3)

5. …, 14, _____, _____, 35, 42, …

(3)

6. Use the digits 4, 5, and 6 once each to write a three-digit
(10) odd number less than 640.

7. Use digits and a comparison symbol to write that two
(Inv. 1) hundred nine is greater than one hundred ninety.

8. It is afternoon. What time will it be 11 12 1
(27) in 6 hours? 10 2

9 3

8 4
76 5

9. Draw a rectangle 3 cm long and 1 cm wide. Then shade
(21, 26) two thirds of it.

10. Find each product:
(28, 29) (a) 2 ¥ 8
(b) 5 ¥ 7 (c) 2 ¥ 7 (d) 5 ¥ 8

126 Saxon Math 5/4 A B
11. In this figure, what type of angle is
(23) angle A?

D C

12. To what number is the arrow pointing?

(Inv. 1)

–20 –10 0 10 20

13. At what temperature does water freeze 17. $5.87
(18) (22) + $2.79
(a) on the Fahrenheit scale?
(b) on the Celsius scale?

14. $83 15. $286 16. 72
(15) – $19 (13) + $387 (15) – 38

18. 19 19. 88 20. 88 21. G
(24) + Q (24) – N (24) – M (24) + 14

46 37 47 47

22. 870 – 470 23. 525 – 521
(14) (14)

24. Change this addition problem to a multiplication problem.
(27, 28) Then find the product on the multiplication table.

8+8+8

25. 1 + 9 + 2 + 8 + 3 + 7 + 4 + 6 + 5 + 10
(1)

26. Which of these does not equal 24?
(28)

A. 3 ¥ 8 B. 4 ¥ 6 C. 2 ¥ 12 D. 8 ¥ 4

27. Name the property of multiplication shown by each of
(28) these examples:

(a) 0 ¥ 50 = 0
(b) 9 ¥ 6 = 6 ¥ 9
(c) 1 ¥ 75 = 75

Lesson 30 127

LESSON

30 Subtracting Three-Digit

Numbers with Regrouping

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Practice splitting the second number to add:

a. 36 + 8 b. 48 + 6 c. 47 + 9
f. 38 + 45 + 200
Review:

d. 67 + 19 + 100 e. 350 + 40 + 200

Patterns:

In some sequences the count from one number to the next
increases. In the sequence below, from 1 to 4 is 3, from 4 to 9 is 5,
and from 9 to 16 is 7. (Notice that the increase itself forms a
sequence.) Continue this sequence to the tenth term, which is 100.

1, 4, 9, 16, …

NEW CONCEPT

We have already learned how to subtract three-digit numbers
without regrouping. In this lesson we will subtract three-digit
numbers with regrouping.

Example 1 Find the difference: $365 – $187 $365
Solution We write the first number on top. We – $187
line up the last digits. We cannot
subtract 7 ones from 5 ones. ?

We exchange 1 ten for 10 ones. Now $ 3 65 15
there are 5 tens and 15 ones. We can – $1 8 7
subtract 7 ones from 15 ones to get
8 ones. 8

We cannot subtract 8 tens from 5 tens, $ 32 165 15
so we exchange 1 hundred for 10 tens. – $1 8 7
Now there are 2 hundreds and 15 tens,
and we can continue subtracting. 78

We subtract 1 hundred from 2 hundreds $ 32 165 15
to finish. The difference is $178. – $1 8 7

$1 7 8

128 Saxon Math 5/4
Example 2 Subtract: $4.10
– $1.12

Solution We subtract pennies, then dimes, and then dollars. We
remember to align the decimal points.

$ 4.10 10 $ 43.110 10 $ 43.110 10
– $ 1.1 2 – $ 1.1 2 – $ 1.1 2

8 .9 8 $ 2.9 8

LESSON PRACTICE b. $4.30 c. 563
– $1.18 – 356
Practice set* Subtract:
a. $365
– $287

d. 240 – 65 e. 459 – 176 f. 157 – 98

MIXED PRACTICE

Problem set 1. The room was full of students when the bell rang. Then
(25) forty-seven students left the room. Twenty-two students

remained. How many students were there when the bell
rang? Write a subtraction pattern and solve the problem.

2. Fifty-six children peered through the window of the pet
(11, 24) shop. After the store owners brought the puppies out,

there were seventy-three children peering through the
window. How many children came to the window after
the puppies were brought out?

3. A nickel is worth 5¢. Gilbert has an even number of nickels
(10) in his pocket. Which of the following could not be the

value of his nickels?

A. 45¢ B. 70¢ C. 20¢

4. It is morning. What time will it be 11 12 1
(27) in 15 minutes? 10 2

9 3

8 4
76 5

Lesson 30 129

5. What is the sixth number in this counting sequence?

(3)

6, 12, 18, …

6. To what number is the arrow pointing?

(Inv. 1)

400 500 600

7. Use a compass to draw a circle with a radius of 1 inch.
(21, 26) Then shade one fourth of the circle.

8. Write 843 in expanded form. Then use words to write the
(7, 16) number.

9. Multiply: (b) 4 ¥ 2 (c) 4 ¥ 5 (d) 6 ¥ 10
(28, 29) (a) 6 ¥ 8

10. Write two addition facts and two subtraction facts using
(6) the numbers 10, 20, and 30.

11. Use a centimeter ruler to measure the rectangle below.
(Inv. 2) (a) How long is the rectangle?

(b) How wide is the rectangle?
(c) What is the perimeter of the rectangle?

12. What type of angle is each angle of a rectangle?
(23)

13. 746 14. $3.86 15. 61 16. $4.86
(30) – 295 (22) + $2.78 (15) – 48 (30) – $2.75

17. 51 + M = 70 18. 86 – A = 43
(24) (24)

19. 25 + Y = 36 20. Q – 24 = 37
(24) (24)

21. (a) Round 89 to the nearest ten.
(20)
(b) Round $8.90 to the nearest dollar.

130 Saxon Math 5/4

22. 25¢ + 25¢ + 25¢ + 25¢
(17)

23. There are 100 cents in a dollar. How many cents are in
(20, 22) half of a dollar?

24. Change this addition problem to a multiplication problem.
(27, 28) Then find the product on the multiplication table.

7+7+7+7+7+7+7

25. 4 + 3 + 8 + 4 + 2 + 5 + 7
(1)

26. Which of these sets of numbers is not an addition/
(6) subtraction fact family?

A. 1, 2, 3 B. 2, 3, 5 C. 2, 4, 6 D. 3, 4, 5

27. Find each product on the multiplication table:
(28) (a) 10 ¥ 10
(b) 11 ¥ 11 (c) 12 ¥ 12

Investigation 3 131

INVESTIGATION 3

Focus on

Multiplication Patterns • Area •
Squares and Square Roots

Multiplication One model of multiplication is a rectangular array. Here we
patterns see an array of 15 stars arranged in three rows and five
columns. This array shows that 3 times 5 equals 15. This
array also shows that 3 and 5 are both factors of 15.

5 columns

3 rows

Refer to this array of X’s to answer problems 1–4 below.

1. How many rows are in the array?
2. How many columns are in the array?
3. How many X’s are in the array?
4. What multiplication fact is illustrated by the array?
Some numbers of objects can be arranged in more than one
array. In problems 5–7 we will work with an array of 12 X’s
that is different from the array we discussed above.
5. Draw an array of 12 X’s arranged in two rows.
6. How many columns of X’s are in the array you drew?
7. What multiplication fact is illustrated by the array you drew?

132 Saxon Math 5/4

Below we show an array of 10 X’s.

8. Which two factors of 10 are shown by this array?
9. Can you draw a rectangular array of ten X’s with three rows?
10. Can you draw a rectangular array of ten X’s with four rows?
11. Can you draw a rectangular array of ten X’s with five rows?
12. Draw an array of X’s arranged in three rows and six

columns. Then write the multiplication fact illustrated by
the array.
13. The chairs in a room were arranged in six rows, with four
chairs in each row. Draw an array that shows this
arrangement, and write the multiplication fact illustrated
by the array.
Area Another model of multiplication is the area model. The area
model is like an array of connected squares. This model
shows that 4 ¥ 6 = 24:

6 squares
on this side

4 squares
on this side

Use 1-cm grid paper to work problems 14–16, 20, and 23–25
below.
14. Outline a 4-cm-by-6-cm rectangle like the one shown above.

How many small squares are in the rectangle?

Investigation 3 133

15. Outline a 3-cm-by-8-cm rectangle. How many small
squares are in the rectangle? What multiplication fact is
illustrated by the rectangle?

16. Outline another rectangle that is made up of 24 squares.
Make this rectangle 2 cm wide. How long is the rectangle?
What multiplication fact is illustrated by the rectangle?

With your finger, trace around the edges of a sheet of paper.
As your finger moves around the paper, it traces the
perimeter of the paper. Now use the palm of your hand to rub
over the surface of the paper. As you do this, your hand
sweeps over the area of the paper. The area is the amount of
surface within the perimeter (boundary) of a flat figure.
17. Use your finger to trace the perimeter of your desktop.
18. Use the palm of your hand to sweep over the area of

your desktop.
We measure the area of a shape by counting the number of
squares of a certain size that are needed to cover its surface.
Here is a square centimeter:

1 cm
1 cm

one square centimeter
(1 sq. cm)

19. How many square centimeters cover the area of this
rectangle?

3 cm
2 cm

20. On 1-cm grid paper, outline a 4-cm-by-3-cm rectangle.
What is the area of the rectangle?

134 Saxon Math 5/4

Here is a square inch:

1 in.

1 in.

one square inch
(1 sq. in.)

21. How many square inches are needed to cover the
rectangle below?

2 in.

2 in.

22. Use your ruler to draw a rectangle 3 in. long and 3 in. wide.
What is the area of the rectangle?

Squares and Some rectangles are squares. A square is a rectangle whose
square roots length and width are equal.

23. On 1-cm grid paper, outline four squares, one each with
the following unit measurements: 1 by 1, 2 by 2, 3 by 3,
and 4 by 4. Write the multiplication fact for each square.

We say that we “square a number” when we multiply a
number by itself. If we square 3, we get 9 because 3 ¥ 3 = 9.
Likewise, 4 squared is 16 because 4 ¥ 4 is 16.
24. What number do we get if we square 6? Outline a square

on grid paper to show the result.
25. What number equals 7 squared? Outline a square on grid

paper to illustrate the answer.

Investigation 3 135

The numbers 1, 4, 9, 16, 25, and so on form a sequence of square
numbers, or perfect squares. Notice that the increase from one
term to the next term forms a sequence of odd numbers.

+3 +5 +7 +9

1, 4, 9, 16, 25, …
26. Find the next five terms in the sequence of square

numbers above.

27. Look back at the multiplication table in Lesson 28. What
pattern do the square numbers make in the table?

To find the square root of a number, we find a number that,
when multiplied by itself, equals the original number. The
square root of 25 is 5 because 5 ¥ 5 = 25. The square root of
36 is 6. A square drawn on grid paper can help us understand
the idea of square roots. When searching for a square root, we
know the number of small squares in all, and we are looking
for the length of a side.

25 squares in all 36 squares in all

5 squares on each side 6 squares on each side
The square root The square root
of 25 is 5. of 36 is 6.

We indicate the square root of a number by using a square
root symbol.

Õ ááá

Square root symbol

We read the symbol as “the square root of.” To read

Õ™∞ = 5

we say, “The square root of twenty-five equals five.”

28. (a) What number equals 9 squared?

(b) What is the square root of 9?

29. Find each square root:

(a) Õ¢ (b) Õ¡§ (c) Õ§¢

30. If the area of a square is 49 square centimeters, how long
is each side of the square?

136 Saxon Math 5/4

LESSON

31 Word Problems

About Comparing

WARM-UP

Facts Practice: 100 Subtraction Facts (Test B)

Mental Math:

Practice splitting the second number to add:

a. 57 + 8 b. 78 + 6 c. 49 + 4
f. 354 + 220 + 18
Review:

d. 300 + 520 + 70 e. 63 + 19 + 200

Patterns:

Here we show four squares. The smallest is made up of 1 small
square. The next three squares are made up of 4, 9, and 16 small
squares. Draw the next two squares in the pattern.

14 9 16

NEW CONCEPT
There are 43 apples in the large basket.

There are 19 apples in the small basket.

When we compare the number of apples in the two baskets,
we see that 43 is greater than 19. To find how much greater
43 is than 19, we subtract.

Larger amount 43
– Smaller amount – 19

Difference 24

Lesson 31 137

As we think about this story, we realize that it is not a “some
went away” story, because nothing went away. This is a
different kind of story. In this story we are comparing two
numbers. One way to compare two numbers is to subtract to
find their difference. We subtract the smaller number from
the larger number. Here we show two ways to write the
subtraction pattern:

Larger
– Smaller

Difference
Larger – smaller = difference

The difference tells us “how many more.” It also tells us
“how many fewer.” There are 24 more apples in the large
basket than there are in the small basket. We can say this
comparison another way. There are 24 fewer apples in the
small basket than there are in the large basket.
Here we show a way to diagram a “larger-smaller-difference”
story. In the diagram we have used the numbers from the
apple story above.

Larger Difference
›‹ ¤›

Smaller
⁄·

Example 1 Forty-two apples is how many more than 13 apples?

Solution To find “how many more,” we use a subtraction pattern. Here
we are comparing the two numbers 42 and 13.

PATTERN PROBLEM
Larger 42 apples
– Smaller – 13 apples
Difference 29 apples

Forty-two apples is 29 apples more than 13 apples.

138 Saxon Math 5/4

Example 2 Seventeen apples is how many fewer than 63 apples?

Solution We are asked to find “how many fewer.” The pattern is the
same as the pattern for finding “how many more.” We use a
subtraction pattern to compare the numbers.

PATTERN PROBLEM
Larger 63 apples
– Smaller – 17 apples
Difference 46 apples

Seventeen apples is 46 apples fewer than 63 apples.

Example 3 Seventeen is how much less than 42?

Solution Problems about numbers that ask “how much less” or “how
much greater” also have a subtraction pattern. This time we
will show the pattern sideways:
PATTERN: Larger – smaller = difference
PROBLEM: 42 – 17 = 25
Seventeen is 25 less than 42.

LESSON PRACTICE

Practice set Write a subtraction pattern for each problem. Then answer
the question.
a. Forty-three is how much greater than twenty-seven?

– b. Mary has 42 peanuts. Frank has 22 peanuts. How many
fewer peanuts does Frank have?

–

– c. Cesar had 53 shells. Juanita had 95 shells. How many
more shells did Juanita have?

–

MIXED PRACTICE

Problem set 1. There were 43 parrots in the tree. Some flew away. Then
(25) there were 27 parrots in the tree. How many parrots flew
–
– away? Write a subtraction pattern and solve the problem.
2. One hundred fifty is how much greater than twenty-three?
1 (31) Write a subtraction pattern and solve the problem.

– 3. Twenty-three apples is how many fewer than seventy-
(31) five apples? Write a subtraction pattern and solve the

– problem.